I think it's 100% appropriate.
Our economy is a war economy as defense spending is a huge chunk of the budget. We've been like that since Korea. On top of that, we conduct military ops around the globe. Not to mention invading and fighting low intensity proxy wars in 3rd world countries for decades that end when it becomes politically untenable with us, literally, running away. Also, we tend to bomb folks semi-regularly. Constant, endless wars.
So, yeah, Dept. of War is appropriate. The slogan should be, "War, it's what we do!"